Question

Break-Even Analysis You invest \(\$ 18,000\) in equipment to make CDs. The CDs can be produced for \(\$ 1.95\) each and will be sold for \(\$ 13.95\) each. How many CDs must you sell to break even?

Short answer

The break-even point, rounded up to the nearest whole number, is approximately 1508 CDs.

Step-by-step solution

Identify the Variables

Identify the knowns which are the fixed cost of the equipment (\$18,000), the cost of making the CDs (\$1.95 each), and the selling price of the CDs (\$13.95 each). Then, identify the unknown, which is the break-even point in the number of CDs. That is the point at which the total revenue equals the total cost, hence the term 'break-even.'

Apply the Break-Even Formula

Apply the formulas: Total Cost = Fixed Costs + Variable Cost • Quantity and Total Revenue = Selling Price • Quantity. Setting the two equal to each other gives us Break-Even Point = Fixed Costs / (Selling Price - Variable Cost).

Solve for Quantity

Substitute the given values into the formula to solve for the quantity of CDs that need to be sold to break even. So, Break-Even Point = $18,000 / ($13.95 - $1.95).

Calculate and Round off to the nearest Whole Number

Calculate the break-even point and, because we cannot sell a fraction of a CD, round up to the nearest whole number if necessary.

Key concepts

Fixed Costs

Fixed costs are expenses that do not change regardless of how much you produce or sell. They remain constant over time and production levels. In the context of our example, the fixed cost is the investment of $18,000 for equipment. This is a one-time cost that needs to be covered before any profits can be made from selling CDs.

• Fixed costs are constant: No matter how many CDs are produced, the $18,000 cost stays the same.
• Understanding fixed costs is crucial in determining the break-even point—knowing how many units you need to sell to cover these costs.
• Common examples include rent, salaries, and insurance, besides equipment costs like in this case.

Variable Cost

Variable costs fluctuate with production volume. They vary directly with the number of units you produce or sell. For the CD production, the variable cost is $1.95 per CD. This means, for every CD made, $1.95 is spent.

• These costs depend on the quantity of production: More CDs mean higher total variable costs.
• Variable costs must be considered in both pricing and production decisions.
• Typical variable costs include materials, labor costs directly tied to production, and utility costs related to manufacturing.

Selling Price

The selling price is the amount of money gained from selling one unit of a product. In our case, each CD is sold for $13.95. The selling price must cover both fixed and variable costs to reach break-even, and ideally, it should also provide a margin for profit.

• The selling price determines potential revenue and profit.
• Setting the correct selling price involves understanding both market demands and cost structure.
• It’s essential for comparing against total costs to find the break-even point.

Revenue

Revenue is the total income generated from selling goods or services. It’s calculated by multiplying the selling price by the number of units sold. For the CDs, the revenue equation is $13.95 (selling price) times the number of CDs sold.

• Revenue is key to analyzing business success; it's not the same as profit.
• Total revenue must surpass total costs for a business to become profitable.
• In break-even analysis, revenue equals costs at the break-even point.

Understanding how revenue interacts with costs helps determine how many CDs need to be sold to achieve profit after covering all expenses, leading towards sustainable business operations.